Problem 44 - Calculus: Early Transcendentals

Question

For \( f(x) = \frac{2}{x} - \frac{1}{\ln x} \) find each of the following limits. (a) \( \displaystyle \lim_{x \to \infty} f(x) \) (b) \( \displaystyle \lim_{x \to 0^+} f(x) \) (c) \( \displaystyle \lim_{x \to 1^-} f(x) \) (d) \( \displaystyle \lim_{x \to 1^+} f(x) \) (e) Use the information from parts (a) - (d) to make a rough sketch of the graph of \( f \).

Step-by-Step Solution

Answer

(a) 0, (b) ∞, (c) ∞, (d) -∞. Graph has vertical asymptote at x=1, horizontal at y=0.

Step 1: Determine limit as x approaches infinity

To find \( \lim_{x \to \infty} f(x) \), evaluate each component separately. As \( x \to \infty \), \( \frac{2}{x} \to 0 \) and \( \frac{1}{\ln x} \to 0 \) because \( \ln x \to \infty \). Therefore, \( \lim_{x \to \infty} \left( \frac{2}{x} - \frac{1}{\ln x} \right) = 0 - 0 = 0 \).

Step 2: Determine limit as x approaches 0 from the right

To find \( \lim_{x \to 0^+} f(x) \), note that as \( x \to 0^+ \), \( \frac{2}{x} \to \infty \). For \( \ln x \to -\infty \), so \( \frac{1}{\ln x} \to 0 \). Thus, \( \lim_{x \to 0^+} \left( \frac{2}{x} - \frac{1}{\ln x} \right) = \infty - 0 = \infty \).

Step 3: Determine limit as x approaches 1 from the left

To find \( \lim_{x \to 1^-} f(x) \), as \( x \to 1^- \), \( \frac{2}{x} \to 2 \) and \( \ln x \to 0^- \). Thus, \( \frac{1}{\ln x} \to -\infty \). Therefore, \( \lim_{x \to 1^-} \left( \frac{2}{x} - \frac{1}{\ln x} \right) = 2 - (-\infty) = \infty \).

Step 4: Determine limit as x approaches 1 from the right

To find \( \lim_{x \to 1^+} f(x) \), note that as \( x \to 1^+ \), \( \frac{2}{x} \to 2 \) and \( \ln x \to 0^+ \). Therefore, \( \frac{1}{\ln x} \to \infty \). Hence, \( \lim_{x \to 1^+} \left( \frac{2}{x} - \frac{1}{\ln x} \right) = 2 - \infty = -\infty \).

Step 5: Use limits to sketch graph

To sketch the graph of \( f(x) \), note that the function approaches 0 as \( x \to \infty \), \( \infty \) as \( x \to 0^+ \), \( \infty \) as \( x \to 1^- \), and \( -\infty \) as \( x \to 1^+ \). These characteristics suggest a vertical asymptote at \( x = 1 \) and a horizontal asymptote at \( y = 0 \). The function rises sharply as \( x \to 0^+ \), decreases smoothly towards y=0 as \( x \to \infty \), and shows discontinuity at \( x = 1 \).

Key Concepts

Asymptotic behavior

Asymptotic behavior in calculus refers to how a function behaves as the independent variable approaches certain values or infinity. For the function \( f(x) = \frac{2}{x} - \frac{1}{\ln x} \), observing its behavior as \( x \) grows large or approaches crucial points like 0 or 1, gives insights into its limits and overall shape. As seen in the original exercise, this function tends to 0 as \( x \to \infty \), meaning its value gets closer to 0 but never quite reaches it. This guides us towards identifying asymptotes which are lines that the graph of a function approaches but never touches. Recognizing such behavior helps in understanding the patterns of change within the function, which is pivotal for sketching or analyzing its graph effectively.

Continuity

Continuity in calculus examines whether a function behaves smoothly and without interruptions. For \( f(x) = \frac{2}{x} - \frac{1}{\ln x} \), the lack of continuity can be observed at \( x = 1 \). The left-hand limit as \( x \to 1^- \) yields \( \infty \) while the right-hand limit as \( x \to 1^+ \) gives \( -\infty \). This drastic jump indicates a discontinuity, which appears visually as a gap or break on a graph where the function does not have a defined value. Observing continuity is essential for understanding how a function behaves over its entire domain and can reveal potential asymptotic vertical lines or breaks where no value technically exists.

Graphical Analysis

Graphical analysis involves visually interpreting the function's behavior based on calculated limits and other properties. For \( f(x) = \frac{2}{x} - \frac{1}{\ln x} \), the limits calculated help in sketching its profile: at \( x \to 0^+ \), the function heads to \( \infty \); as \( x \to \infty \), it approaches 0. Additionally, the function behaves erratically around \( x = 1 \), where it experiences discontinuity. These behaviors together help sketch its general shape. Graphically plotting such observations can help students better predict how a graph will look and what features, like peaks or discontinuities, they should expect. It presents a more intuitive understanding of mathematical relationships conveyed via the function.

Horizontal and Vertical Asymptotes

Identifying asymptotes provides a framework around which functions can be analyzed. For this function, the process of determining horizontal and vertical asymptotes is driven by finding which values the function approaches but never quite reaches. A horizontal asymptote is noticed at \( y = 0 \) because as \( x \to \infty \), \( f(x) \to 0 \). This implies the function flattens or levels off at this line as \( x \) increases indefinitely. Meanwhile, a vertical asymptote is found at \( x = 1 \) due to the sharp directional changes in the limits as \( x \to 1^- \) and \( x \to 1^+ \). Understanding these asymptotes is crucial for drawing functions correctly and predicting behavior like potential infinite increases or drops.